Distinguish scalar and vector quantities and apply vector addition, subtraction and resolution into perpendicular components in one and two dimensions

What this dot point is asking

VCAA wants you to use the scalar/vector distinction with precision and to resolve and combine vectors using components.

Scalars and vectors

Vector addition

Graphical (head-to-tail). Place tail of second at head of first; resultant runs from tail of first to head of last.

Components. For a vector of magnitude $v$ at angle $\theta$ above the horizontal:

Sum $x$-components, sum $y$-components, recombine:

Vector subtraction

$\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$. Reverse $\vec{b}$ and add. This is the key step for $\Delta \vec{v} = \vec{v}_f - \vec{v}_i$ in collisions and uniform circular motion.

Worked example

A car moves east at $20$ m s$^{-1}$ then turns to move north at $20$ m s$^{-1}$. Change in velocity:

$\Delta \vec{v} = 20 \hat\jmath - 20 \hat\imath$. Magnitude $\sqrt{20^2 + 20^2} = 28.3$ m s$^{-1}$.

Direction: $45°$ north of west.

The speed did not change but the velocity did. This is the source of centripetal acceleration in circular motion.

Common traps

Adding magnitudes of perpendicular vectors

$3$ m east plus $4$ m north is $5$ m displacement, not $7$ m.

Confusing speed and velocity

Speed is the magnitude of velocity. A car turning at constant speed has changing velocity.

Mixing degrees and radians on the calculator

VCE Physics expects degrees unless specified.

In one sentence

Scalars have magnitude only; vectors have magnitude and direction (displacement, velocity, force, momentum) and are combined by head-to-tail addition or by resolving into perpendicular components, with subtraction performed by reversing the subtracted vector.

Examples in context

Example 1. Tullamarine runway crosswind component. A pilot lands at Tullamarine runway 27 (heading $270^\circ$, due west) with a wind from $210^\circ$ at $25$ knots. The angle between the wind vector and runway direction is $60^\circ$. Crosswind component (perpendicular to runway) is $25 \sin 60^\circ = 21.7$ kt; headwind component is $25 \cos 60^\circ = 12.5$ kt. The crosswind component approaches the A330's $30$ kt limit, so the captain may opt to use runway 34 (heading $340^\circ$) where the same wind gives crosswind $25 \sin 130^\circ = 19.2$ kt and tailwind $25 \cos 130^\circ = -16.1$ kt (a tailwind). Vector resolution is the routine quantitative tool for runway selection.

Example 2. Yarra River rowing crew velocity vectors. A Melbourne Mercantile rowing crew accelerates upstream on the Yarra at $4$ m s$^{-1}$ relative to water; the river flows downstream at $1.2$ m s$^{-1}$. Relative to the bank, ground speed is $4 - 1.2 = 2.8$ m s$^{-1}$ upstream. Now consider a ferry crossing at $3$ m s$^{-1}$ perpendicular to a $1.2$ m s$^{-1}$ current: the ground velocity has magnitude $\sqrt{3^2 + 1.2^2} = 3.23$ m s$^{-1}$ at angle $\arctan(1.2/3) = 21.8^\circ$ off the perpendicular. Both examples illustrate vector addition by component, used by coxswains to set heading offsets that produce a straight ground track.

Try this

Q1. Distinguish scalar from vector quantity, with one example of each. [2 marks]

• Cue. Scalar: magnitude only (mass, time, energy). Vector: magnitude and direction (velocity, force, displacement).

Q2. Two forces act on a point: $30$ N east and $40$ N north. Calculate (a) the resultant magnitude, and (b) the angle measured east of north. [4 marks]

• Cue. (a) $\sqrt{900 + 1600} = 50$ N. (b) $\arctan(30/40) = 36.9^\circ$.

Q3. Refer to the Tullamarine crosswind. (a) Resolve a $25$ kt wind from $210^\circ$ into headwind and crosswind components for runway 27. (b) Calculate components for runway 34. (c) Recommend which runway has the lower crosswind. [3+3+1 marks]

• Cue. (a) Headwind $12.5$ kt, crosswind $21.7$ kt. (b) Headwind $-16.1$ kt (tailwind), crosswind $19.2$ kt. (c) Runway 34 has lower crosswind but tailwind; runway 27 still preferred.